Specialist Mathematics · Past QCAA questions All simple familiar questions
2023 · Paper 2 · Simple familiar

Question 12

[7 marks] Technology-active
Topic guidesPremiumUnit 3 Topic 1.1 · The Unit 2 basics you needUnit 3 Topic 1.1 · De Moivre’s theorem
Game plansPremium1.1 Converting between Cartesian and polar form1.2 Choosing the form for the arithmetic
The question

Consider the complex number $z=-3+2i$.

a)

Determine $z^3$ using the binomial theorem. Leave your answer in the form $a+bi$, where $a,b\in\mathbb{R}$.

[2 marks] b)

Convert $z$ into the form of $r\operatorname{cis}(\theta)$, where $-\pi<\theta\le\pi$.

[1 mark] c)

Use the result from Question 12b) to determine $z^3$ using De Moivre’s theorem. Leave your answer in the form of $r\operatorname{cis}(\theta)$, where $-\pi<\theta\le\pi$.

[2 marks] d)

Evaluate the reasonableness of your results from Questions 12a) and 12c), noting that the two methods to determine $z^3$ should produce the same result.

[2 marks]
Watch the situation first

Triple the angle and go round more than once

Cubing $z$ triples its argument. Watch where the tripled angle lands, then check it against the answer from the binomial theorem.

z = −3 + 2i is in the second quadrant Its argument is between π/2 and π, so θ ≈ 2.55 De Moivre: cube the modulus and triple the argument 3θ ≈ 7.66 is more than π, so subtract 2π z3 ≈ 46.87 cis(1.38), drawn shorter so it fits The binomial answer 9 + 46i points the same way ReIm θ z = −3 + 2i z3 946 r = √13 ≈ 3.61 θ = π − tan−1(2/3) ≈ 2.55 r3 ≈ 46.87, 3θ ≈ 7.66 7.66 − 2π ≈ 1.38 z3 ≈ 46.87 cis(1.38) √(92 + 462) ≈ 46.87 tan−1(46/9) ≈ 1.38

The arrow for $z^3$ is drawn about a quarter of its true length so it fits. Its direction is exact.

Now for the mathematics

See how rounding moves the answer

Part d) asks why the two answers might not match exactly. Choose how much to round the modulus and the argument before using De Moivre’s theorem, and see how far the result lands from $9+46i$.

024681012 444648 Zoomed in near 9 + 46i 9 + 46i rounded
Round the modulus to
Round the argument to

With nothing rounded, De Moivre’s theorem gives exactly $9+46i$. On your GDC, store the modulus and argument and use the stored values to get this.

The rounded answer is within $0.6$ of $9+46i$. That is a small difference caused by rounding, so the two methods are consistent.

The answer has drifted a long way. Rounding the argument to one decimal place is the main cause, because cubing triples the angle error and the modulus is about $47$, so a small angle error moves the point a long way. Keep more decimal places, or use stored values.

QCAA marking guide · 7 marks

Work through the solution one mark at a time

Try each step yourself before you reveal it. The steps follow the QCAA sample response.

Step 1 · part a)1 mark

Expand with the binomial theorem, using $a=-3$ and $b=2i$. The coefficients for a cube are $1,3,3,1$.

$$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$

$$z^3=(-3)^3+3(-3)^2(2i)+3(-3)(2i)^2+(2i)^3$$

This working has to be shown. Typing $(-3+2i)^3$ into your GDC doesn’t earn this mark.

Marker: correctly uses the binomial theorem

Step 2 · part a)1 mark

Simplify each term using $i^2=-1$ and $i^3=-i$, then collect the real and imaginary parts.

$$=-27+54i+36-8i$$

$z^3=9+46i$

Marker: expresses $z^3$ in the form of $a+bi$

Step 3 · part b)1 mark

Find the modulus, then the argument. $z$ is in the second quadrant, so the argument is between $\frac{\pi}{2}$ and $\pi$. Your GDC can also convert it directly, in radians.

$$r=\sqrt{(-3)^2+2^2}=\sqrt{13}\approx3.61\qquad\theta=\pi-\tan^{-1}\left(\frac23\right)\approx2.55$$

$z\approx3.61\operatorname{cis}(2.55)$

The QCAA also accept $\sqrt{13}\operatorname{cis}(2.6)$.

Marker: correctly converts $z$ into $r\operatorname{cis}(\theta)$ form where $-\pi<\theta\le\pi$

Step 4 · part c)1 mark

De Moivre’s theorem cubes the modulus and triples the argument. Use the stored values from part b) to avoid rounding errors.

$$\big(r\operatorname{cis}\theta\big)^n=r^n\operatorname{cis}(n\theta)$$

$$z^3=\left(\sqrt{13}\right)^3\operatorname{cis}(3\times2.5536)\approx46.87\operatorname{cis}(7.66)$$

The working that shows De Moivre’s theorem must be there to earn this mark.

Marker: uses De Moivre’s theorem to calculate $z^3$

Step 5 · part c)1 mark

The argument $7.66$ is outside $-\pi<\theta\le\pi$. Subtract $2\pi$ to bring it back into the range. That is the same direction, one full turn less.

$$7.66-2\pi\approx1.38$$

$z^3\approx46.87\operatorname{cis}(1.38)$

The QCAA also accept $\left(\sqrt{13}\right)^3\operatorname{cis}(1.38)$ and $47\operatorname{cis}(1.4)$.

Marker: determines $z^3$ in the form of $r\operatorname{cis}(\theta)$ where $-\pi<\theta\le\pi$

Step 6 · part d)1 mark

To compare the two answers, write them in the same form. Convert the polar answer back to Cartesian form.

$$46.87\big(\cos(1.38)+i\sin(1.38)\big)\approx8.89+46.02i$$

You could instead convert $9+46i$ to polar form: $\sqrt{9^2+46^2}\approx46.87$ and $\tan^{-1}\left(\frac{46}{9}\right)\approx1.38$. Either way earns the mark.

Marker: shows mathematical reasoning to convert from $r\operatorname{cis}(\theta)$ form to $a+bi$ form

Step 7 · part d)1 mark

Make a decision and give the reason for any difference.

The two methods give approximately the same value of $z^3$. The small difference comes from rounding in the earlier calculations, so both results are reasonable.

If you used exact or stored values and got exactly $9+46i$, say that the results match exactly. That decision also earns the mark.

Marker: states a decision regarding the reasonableness

Putting it all together

Each phrase of the question gave you something

“using the binomial theorem”
The expansion has to be written out. A GDC answer alone earns no marks here.
“where $-\pi<\theta\le\pi$”
This is the principal argument. In part c) it tells you to subtract $2\pi$ from $7.66$.
“Use the result from Question 12b)”
Apply De Moivre’s theorem to your polar form, and show it.
“Evaluate the reasonableness”
Put both answers in the same form, compare them, then state a decision with a reason.
What makes this simple familiar

Each part is a routine complex number skill, and the question tells you which method to use every time. Part d) is the standard “compare the two methods” check.

If I had to guess where you might lose marks, it would be leaving the argument as $7.66$ in part c), which is outside the range the question asks for. In part d), saying “they are the same” isn’t enough. Convert one answer, compare, and explain the small difference.

Only mark this done when you could do it without help. Reading the solution does not count.

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Question wording and marking-guide steps are from the 2023 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.

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